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Say you have two surfaces of genus 2, say $X$ and $Y$ and you want to attach them via homotopy attaching maps $f$ along their waist curves. Then what will the fundamental group of the adjunction space $X\bigcup_fY$ be?

Any Hints on how to proceed?


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Which four circles are you seeing? It seems to me you'd get a genus 4 surface out of this, not genus 3. – Kevin Carlson Oct 26 '12 at 3:51
Now, that i think about it, i'm not actually sure. But, it seems more what you were saying, that it will be a surface of genus 4. So, in this case then it is homeomorphic to a connected sum of four tori, so the fundamental group would be simply $\pi_1(\mathbb{T})\star\pi_1(\mathbb{T})\star\pi_1(\mathbb{T})\star\pi_1(\mathbb‌​{T})=\mathbb{Z}^2\star\mathbb{Z}^2\star\mathbb{Z}^2\star\mathbb{Z}^2.$ Right? With $\star$ I have denoted the free product. – susan Oct 26 '12 at 4:38
A presentation for the fundamental group would then be: $\langle\alpha_1,\beta_1,\alpha_2,\beta_2,\alpha_3,\beta_3,\alpha_4,\beta_4: \alpha_1 \beta_1\alpha_1^{-1}\beta_1^{-1}...\alpha_4\beta_4\alpha_4^{-1}\beta_{4}^{-1}\ra‌​ngle$ – susan Oct 26 '12 at 4:50
that would do it. As I say, I'm not sure of this analysis. – Kevin Carlson Oct 26 '12 at 4:54
Yeah, I'm not so sure myself either...let's see if someone else gives some more insight??? – susan Oct 26 '12 at 5:12

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